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Gauge independent Wigner function approach to spontaneous emission and to quantum kinetic equations
Physica 130A (1985) 565-586 North-Holland, Amsterdam
GAUGE
INDEPENDENT
SPONTANEOUS
WIGNER FUNCTION APPROACH
EMISSION AND TO QUANTUM
KINETIC
TO
EQUATIONS
Piotr BADZIAG of Physics, University of South Africa, P 0 Box 392, Pretoria, South Africa
Deparhnent
Received 5 July 1984 Revised 4 October 1984
Equations for the evolution of the gauge independent analogue of the Wigner function are derived and applied to the description of a many charged particle system. The self-consistent field approximation is studied more closely. Possible applications of the formalism are illustrated by the calculations of dispersion relations for elementary excitations of fermion as well as boson systems. A possibility of application of the formalism in quantum optics is also investigated.
1. Introduction
The Weyl-Wigner formulation of quantum mechanics’,*) has been widely used in different fields of physics. There are well-known Wigner formulations of scattering theory3*4), relativistic quantum mechanics5*6) and quantum kinetic theory7-9) as well as some approaches via the Wigner representation to quantum electrodynamics”) and quantum field theory”). Although the Wigner representation is not commonly used in above-mentioned fields because of the complexity of the equations, in most of the applications it is very useful in the investigations of the long wave phenomena directly connected to the semiclassical limit. In this limit the Wigner function becomes the classical phase space density, whereas describing charged particles in an electromagnetic field neither Wigner function nor phase space density are real objects,
since they are not gauge independent.One
can slightly
change
the
definition of the Wigner function to make it gauge independent and to give it an interpretation of the quantum analogue of the classical density in the position-kinetic momenta space’,“). Th e main purpose of this paper is to apply the gauge independent Wigner function formalism to the kinetic theory of charged particle gas as well as to the description of the interaction between atoms and quantized electromagnetic field. When dealing with kinetic equations we restrict ourselves to the spin f and spinless particles and finally 0378-4371/85/$03.30 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
P. BADZIAG
566
concentrate on the self-consistent field approximation. gives elementary definitions, examples and theorems representation’).
In section
charged particle We also present
3 we redefine
Section connected
the Wigner
2 of this paper to the Wigner
function
describing
a
in an electro-magnetic field to make it gauge independent**“). some properties of the gauge independent Wigner function.
This section
introduces
also the generalization
ning particle
case. In section
4 we begin
of the formalism
with the description
for the spin-
of many
particle
systems: introduce a concept of reduced gauge independent Wigner functions and list some properties of them. Section 5 gives the discussion of the quantum Vlasov approximation and comparison of the approximation with the HartreeFock one. Sections 6 and 7 give a discussion of the linearized Vlasov equation describing fermion and boson systems, respectively. The important role of the coherent states13) in the case of Bose particles becomes here visible’4X’5). Our approach to derive the Landau criterion for superfluidity can be in this context compared with the one of Gross”); however, do not need to assume that the excitations
to derive the proper formula, we are long wave ones, what Gross
actually needed (linearization not only in the density variations 6p but also in the variations of the phase of the wave function 8s). Finally, in section 8 we generalize the formalism to the case when the electromagnetic field is quantized and show that this generalization enables a straightforward description of atoms interaction with a quantum electromagnetic field. Some concluding remarks are presented in section 9.
2. Wigner function The Wigner
function
matrix
in coordinate
particle
its Wigner
is usually
defined
representation. representation
as a Fourier
When is defined
density
transform matrix
of the density
fi describes
one
as follows:
(2.1) The function defined in this way is the quantum analogue of the classical phase space density. To justify this statement let us enumerate some of the properties of the Wigner function”). Expressions for the coordinate space probability density, momentum space probability density and current density are given by classical relations:
i)
p(R, f) = j
d”pf,(R
P;
1)2 0,
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
ii>
iii)
P(P,t) = j d3Rfw(R P; t) 2 0 , j (R, t) = j dp tf,(R,
p; t) .
If 8 is the operator which represents an observable, then the expectation of 8 in the state described by fw( .) is given by
iv)
561
(6) = 1d3pdZRO,(R, p)f,(R,
value
p; 0.
Here O&t, P; t) =
1d3r (R -
kr18(R + it-) exp (ip
- r)
is the Weyl transform of the 8 operator and is closely related to the classical observable. One can easily show for example that
W%(R
P)
= f(p),
[f(@l,(RP) =f(R).
Of course one cannot prove that same for arbitrary f(k, $), because k and a do not commute. Although the Wigner function itself is real, it is not the density in the classical meaning, since it is not non-negative. This is connected with the uncertainty principle, so that the quantum analogue of the positive definiteness of the classical distribution function reads:
v)
(27r~rp)-~ 1 d3p d3Rfw(R, p) exp [ - w
- (RiPF)2]
2 0.
To prove (v)‘“) note that for two density matrices
(*I In the Wigner representation
(*) reads:
568
P. BADZIAG
Thus,
to obtain
(v) we have only to write
where
h2
(y2=-
4p2. f2 defined above is the Wigner function packet given by the wave function
i+b(R) = (27@2)-3’2exp
The Wigner representation quite similar to the classical
[
- (R4B?)2]
which
describes
the Gaussian
wave
exp (- i pO - R)
of the Liouville-von one. For a particle
Neuman in external
equation potential
has a form field Q(R)
this reads
(2.2) Here F(R)
= -V@(R)
1
112
d3r d3q
=
(2n-h)3
exp
dA F(R
[-ir*(P-cl)
+ Ar)
l/2
i =-Ih
d3r d2q (277-h)3 exp
I
-~r*(p-q)].[@(R+~r)-@(R-;r)]fw(R,q).
stfw(R.
q)
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
569
To make the semiclassical limit of the integral operation &[ F(R)] more visible one can write it in the form of the formal series of the differential operators:
One can easily see now that in the limit when F(R) is a slowly varying function eq. (2.2) gives the classical Liouville equation. One can also see that in the case of F being linearly varying (constant force, harmonic oscillator) (2.2) is exactly the same as its classical counterpart.
3. Charged particle in an electromagnetic
field
If we consider a charged particle in an electromagnetic field, eq. (2.2) is no longer so transparent. One of the reasons for it is that &(R,p; t), just like the classical phase space distribution function, depends on the choice of gauge of electromagnetic potentials. Therefore in classical theory one often uses the position-velocity space distribution instead of the phase space one. Following this idea we modify the definition of the Wigner function to make it a quantum analogue of the classical position-kinetic momentum density. There are of course many different possibilities for such a modification”). To make our choice more unique let us rewrite definition (2.1) in the form:
(3.1)
Simple, but a little tedious algebra shows that (3.1) and (2.1) are indeed equivalent. By substituting canonical momentum operator @ by the kinetic momentum one, P - (e/c)A in (3.1) we get (compare refs. 8 and 11). l/2
exp{-ir.[p+E
1
dAA(R+hr,t)]]
-l/Z
xp(R+;r,R-;r;
t).
(3.2)
570
P. BADZIAG
One
can easily
and
has properties
position
verify
and kinetic
any observable
that the function
of the quantum momentum
defined analogue
space.
by (3.2) is gauge of the classical
In particular
independent density
the expectation
in the value
of
8 is given by:
(6) = j- d3Rd3pO(R, p)f(R P)
(3.3)
1
where O(R,p) is the Weyl transform of the 0 operator modified in a similar way to the Wigner function. For the one particle operator it reads:
O(R,p)=ld’r(R-;r@jR+;r) l/2
xexp(kr*[p+:
1
dAA(R+hr,f)]].
(3.4)
-l/2
Thus, for instance, reads :
the probability
density
of finding
a particle
in d3R around
AR, 0 = (IJWRI)= j- d3pf(RP; f) ; the current
R
(3.5)
density:
(3.6)
Different coordinates of the kinetic momentum operator do not commute, so that one cannot expect that property (v) of the Wigner function transforms directly into the gauge independent function. There is, however, an analogue of that property which, at least in the case when the magnetic field is uniform, has a direct and transparent form -namely: for &P+P, 1 f&R,,
~0; t) = ~2Tap~~
x exp _
I
1
d3p d3R
f (R, p; t) exp
-
(R - R,)* w2
b
-PO
+
I.
(42cP x CR- Ro)12 , 2CX2
1
o
(3.7)
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
571
The proof of (3.7) is just the same as that of (v) in section 2. It is only to be noted that the pure state described in the Coulomb gauge by the wave function
JI(R)
=
Q+*)-~‘*
exp
in the gauge independent
-
(Rip?‘2 +f R * (p.+5 B x R,,)]
Wigner representation
f2M P) = (27~Lyp)-~ exp{ -
(R - Ro)* _ [P -PO
is given by +
W2c)Bx (R -
2p2
2cu2
Ro)l*
,
I
where (y*=-
h2 4p* *
It is worthwhile to point out, that f*(a) d escribes a state in which position and kinetic momentum coordinates perpendicular to B are uncertain up to 2 u ,l
-p*,
e2B2p2
u;,=a*+4cZ.
3.1. Evolution of the gauge independent Wigner function It is just a matter of a simple but quite tedious algebra to show that the Liouvelle-von Neuman equation in the gauge independent Wigner representation reads: l$tL11_P+e(ti[E(R,t)]+fL;,x~[B(R,t)])+}f(R,p;t)=0.
Here fi[F]
is described in section 2; B,f=$[pf(R,p)+~~~
7 dA exp(fr.q)[B(R-hr)+hr]) -l/2
xf(R,p-4)
(3.8)
572
P. BADZIAG
In the limit when E and B are slowly varying for the evolution of the probability space. Here, like in the orthodox
density Wigner
forces, (3.8) is the same as its classical and B is uniform. 3.2. Spinning
(3.8) gives the classical in coordinate-kinetic representation and
limit when E changes
equation
momentum the potential
linearily
in space
particle
Eq. (3.8) describes the particle possesses nonzero
evolution of the spinless spin the density operator
particle only. in coordinate
When the represen-
tation is the matrix function paP(R,, R,;t).where (Y.p are spin indices. This indicates that instead of introducing only one Wigner function we have to introduce the matrix of functions: faa( R,p; t). This makes the whole formalism more complicated; however, at least for spin i particles. all linear independent functions have a clear physical interpretation, namely
describes
the quantum
analogue
of probability
density.
with u the Pauli matrices, describes 2/h times the spin density. For the time being we restrict ourselves to this simple case. In this case, instead of (3.8) we have
+ ~~[B(R,t)](6.~~.s(R,P:t)=O.
(3.9a)
(3.9b)
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
573
One can easily see that when B is uniform integration of (3.9b) over the momentum space gives the well known equation for spin precession:
(3.10)
~s.(Qf)+vj~j;(R,l)+~EiPBISI(Qf)=O. Here si(R, r> = 1 d3Psi(R, p; t) describes the space distribution
of spin
Tji(R,t)=Idip~p,si(R,p;t); which describes the spin current.
4. Many particle system To describe a many particle system one has to work with the reduced density matrices. The same applies to the Wigner formalism. We define the one particle reduced gauge independent Wigner function for spin zero particles as follows: l/2
3 (:exp[-ii*[p+: h(R~Wj&y
1 d,iA(R+Ar,l)]} -l/2
x I&R -
;r,t)t,i(R + ;r):).
(4.1)
Here 4 are the field operators, :( ): is the symbol of normal ordering and ( ) is the symbol of the expectation value. The generalisation of (4.1) on the spinning particles is straightforward. We only have to add spin indices to fi, which now is the matrix fld and to 4’ and 4 which now become 4: and &. For spin l particles instead off,& we can choose the following linear independent combinations of these functions:
f,(R,p; O= ~fl,za(R~; 6, n
(4.2a)
P BADZIAG
574
s,(R
Pi
t) = c u,,f,,(Rp;t)
(4.2b)
nb
Here &‘s are the Pauli matrices. The many particle reduced defined
in an analogous
fzabcAR,, PI; R,,
gauge
independent
way, i.e. two particle
Wigner
functions
functions
are
are
PS t) 112
1 dAA(R,+Ar,,l)]}
=]$$$(:exp{-iir,-[ktf
-112 x I&R, - ;r,, t)&(R, Commutation relations between conditions for f2 functions: f2,,‘&R,,
~1;
R,,
~2;
f> =
- 1 ?r2, t)x ~~(R,+~r,,t)~b(R,+~r,, field operators
fwati&
P2; RI,
yield
the following
t): symmetry
(4.4a)
PI; f, 3
+w,.(R,--R,-~r,-fr2)+w2~(RZ-R,-~rl-~r2)+~
c 1 d* S - Bk
=: *&2f2abcd(R,t PI; R,, P2; t). Here
0
is the surface
f)] ]
(4.4b)
of the quadrangle
bounded
by the sectors
between
the
points R,-ir,,
R,+ir,,
R,+ir,,
R,-kr,.
The order of these points and the right-hand screw rule give direction of the surface, the element of which is denoted by d*S.
5. Evolution
of f,
Particles
interact
with each other,
so that the evolution
equation
the
for
positive
fi
is no
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
longer as simple as (3.8) or (3.9). If the interaction is via the potential with potential U(jRI), we get the following equation for f,:
d3R’d3p’lk[VU(~R-R’~)]~$f,,,(R,p;R’,~’;~).
=-
515
force
(5.1)
Here p denotes the gyromagnetic ratio, ii are the spin divided by h operators. One can rewrite the right-hand side of (5.1) in a more evident form: d3q d3r d3R’ d p (2d)3 exp 3
&
= f
,
’
x[U(jR-R’+;tl)-U(IR-R’-;rl)]f2dcc(R,q;R’,p’;f).
(5la)
Equations for f, involve f3 and so on, and thus we obtain the quantum analogue of classical BBGKY hierarchy. This is, however, still a hierarchy of classical character in a sense that the electromagnetic field is considered here as a given c-number field. If it is not the case, instead of products of the type
we
have
:@(R, th(R, where k ordering. going to imation)
P;
t): ,
and f& are quantum mechanical operators and :( ): denotes normal We will return to this problem in section 8 and for the present we are investigate the self-consistent field approximation (Vlasov approxof eq. (5.1).
576
P. BADZIAG
The
classical
Vlasov
q; t)*flcc(R’,p’;
f,,(R,
does not, however, quantum
theory
approximation t) instead
preserve
is obtained,
when
in (5.la)
one
puts
of fzabcc(R, q; R’, p’; t). This approximation
the proper
symmetry
of fi. Because
of this, in the
one has to put rather (5.2)
Here
the upper
proper
sign refers
self-consistent
to bosons
and the lower
field approximation
fmcc(R 4; R’, P'; f)==#+ &f,,(R’,p’;
for bosons t)f,JR,
to fermions.
Actually
the
is rather q;
r) .
In section 7 we discuss this problem in more detail. To see the quantum corrections to the classical Vlasov equation let us examine the quantum Vlasov approximation of eq. (5.1). We will do it separately for fermions and separately for bosons since the natures of the ground state for both types of particles are different, ergo the different types of approximation
are useful.
6. Elementary
excitations - fermions
For the sake of simplicity we restrict ourselves to spin i particles. In this case we can rewrite the quantum Vlasov approximation of (5.1) in the form of coupled equations for f, and s, defined by (4.2a) and (4.2b), respectively:
(6.1 a)
(&lb)
Here
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
d3q d3r
R,(R,p;r)=$/d3R’d3p’----
Lr*(p-q)][U(lR-R’+;rl) h
(27rh)3 exp
h)f,(R
WIR - R’ - ;rl)l(l -
4;
t)f,(R’, P’; 9, &p-q)
-
U(l R - R’ - ;rj)](l
511
I
[~(IR-~f+;rl)
- &)s,(R, q; t)f,(R’, P’; t) .
In the case of the magnetic field being not too strong, i.e. eiiB -4~: C
for low temperatures,
or eflB
-4
C
2mk,T
for the high temperature
limit,
we can neglect the magnetic field in the definition (4.4b) of the p,* operation, so that in this case we can write
B,,[~,(R’,P’)~(R,‘I)I=~~~~~~(~[~.(R-R’)-(P-~).~IJ R+R’-r
p’+q+T
2
)
2
R+R’+r >fi (
2
p+q-7 ’
2
)
(6.2a)
~~~[f,(R’,p’)s,(R,q)l~~~~exp{~~~.(R-R’)-(~’-q).rl} R+R’-r 2 +
(s1*f1>-
p’+q+T )
%,f,
R+R’+r 2
2
--, s, x Sl> . 1
p+q-7 ’
2
>
(6.2b)
This approximation makes (6.la) and (6.lb) even simpler than their gauge dependent analogue, known e.g. from Silin’s or Balescu’s textbooks. The
578
P. BADZIAG
information equations
contained
our approach wave
in (6.1) is of course
for the phase
space Wigner
the quantities
limit,
a straightforward,
the absence
of the magnetic
the same as the one contained
function.
that we are dealing classical,
gauge
The main with have, independent
field our equations
difference
in the
is that in
at least in the long interpretation.
are the same as those
In
for the
phase space Wigner function and after linearization describe the longitudinal plasma oscillations (when fO is the Maxwell distribution) as well as the excitations of the Fermi liquid (when fO is the Fermi-Dirac distribution and describes quasiparticles rather than particles). The crucial role of the exchange term becomes apparent in the later case. We are, however, not going to proceed along this line, because the results are well known and the reader can find them in textbooks like those by Silin or by Balescu’).
7. Elementary
excitations - bosons
In this section we are going to investigate the dispersion relations for the elementary excitations in a system of weakly interacting Bose particles in the low temperature limit. For the sake of simplicity we restrict ourselves to the case of spinless particles and no external magnetic field present. The quantum Vlasov approximation of eq. (5.1) now reads:
x ;[U(jR
-R’+;rj)-
U/(/R - R’-;rl)](l
+ &f,(R,
q; f)f,(R’,p’;
t).
(7.1) To derive
(7.1) let us have
a look
at the
derivation
of the
Hartree-Fock
approximation. The basic assumption which leads to this approximation all the particles (quasiparticles) are in different one-particle states, N-particle state is of the form
I$,)= ii;, . . . a;p>, where
all k,, . . . , k,
is that i.e. an
(7.2)
are different.
Here I@) is the vacuum state and (~2:~) are creation operators the states {ki}. Function fi( * ) is the Fourier transform of the expectation product:
of particles value
in
of the
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
The expectation
519
value of (7.3) in the state (7.2) is
The last equality leads directly to
When we deal with Bose particles, states like the one described by (7.2) are of no value, because in this case many particles can occupy the same quantum state. Instead of the states of the type (7.2) the coherent states are distinguished now (at least in the problems connected to superfluidity). Coherent states are the eigenstates of anihilation operators, i.e. (7.5) This leads to
We choose the last form, because it preserves the proper symmetry of the interaction term in the evolution equation. The last equality in (7.6) is equivalent to
f*( - ) =
a+ kM(
-)f*( - )
7
which we use in (7.1). For spinless particles and no magnetic field we have
(7.7)
P. BADZIAG
580
Xf,(;[R+ RIf we now collect
r],;[p’+
q + ~l)f,(;[R+
R’+ r],;[p’+q-
T]).
(7.8)
(7.1) and (7.8), then substitute
f,(R, P; t) = A,(P) + V(R> P; t) and leave only the linear
X
the spin zero analogue
of (6.3):
exp [-~(R-R’).(p-q)j-ti(2ip-qj)~~p{$R-R’)~(p-q)]j
x { Cf(p’Pf(R +
terms in Sf, we obtain
d3r d37 (2d)”
exp
t) + fo(qPf(R’, p’; f)l
4;
{&(R-R’)-(p’-q)v]} h
x [f&p’+ q + dPf(;[R + R’+ X 6f(i[R
+ R’-
r], l[p’+
q +
T];
r],f[p’+ r)]}.
q -
71;t>+fo&~'+
9-
71) (7.9)
When
6f(R,P; f>= exp {f (k - R -
4}f,(p)
eq. (7.9) gives
i
k+pJm -w+l
=
1 d3q ol(l~ -
ql)[fo(q+ ;k) -fob - %)I\fdd
;[fo(p+ ;k)-fob - ;kl j d3q[~#I)+
When the interaction most of the particles
ol(lp
-
ql)lfk(q).
is weak and repulsive, in the ground are condensated, so we can write
(7.10)
state of the system
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
581
h(P) = %6(P). This gives the final form of the equation for fk(p):
= ;~,P(P + $4 - S(P- ;k)lI d3q[~(IW + ~I(IP- ~l)ltX~).
(7.11)
Because of the delta functions on the right-hand side of (7.11) we can get get dispersion relation from this equation without making any additional approximations. We can rewrite (7.11) in the form
= ;@(P
-
;@(P
+
I
0th + ;kl)]fk(q) ;k) d3q[ fiI(lkl)+
-;k)I d3q[fil(lkl)+ oI(lq- ~kl>l.h(4)
(7.12)
3
or ;r@(p ‘(‘)
+ ;k)
= -k*/(2m) - w + in,,[ l_?(O)- e(jkl)] ;n,S(p
xk
- ;k)
- k2/(2m) - o + $n,,[o(lkj) - a(O)] yk ’
(7.12’)
where by X, and Yk we have denoted the following integrals:
xk
=
yk = Integration
d3q[~l(lkl)+fil(bl+;kl)l_fi(‘d 3
d3q[~(lkl) + ol(i’I- ;kl)lf,(‘d. over d3p of the both sides of (7.12’) multiplied by
1) {o(M) + ti(I, + ;kl)L
2)
@l(lkl) + o(lp - %I)1
gives the linear and homogeneous set of two equations solvability condition for this set reads:
for X, and Yk. The
P. BADZIAG
582
w2
_
_
k2
I 24W*
(2m)’
2m
(7.13)
’
where v(k) = n,,~(~k~) The relation (7.13) obtained from the proper self-consistent field proximation (7.1) is just the same as the one obtained from diagonalization the many particle Hamiltonian by the Bogolubov transformation method e.g. ref. 18) and because of the coherent states, which are the basis of self-consistent field approximation, it is related to the same result obtained GrossIS) from the hydrodynamical form of quantum mechanics.
8. Atoms in a quantized
electromagnetic
apof (of our by
field
So far we have been investigating systems, where the electromagnetic field was a c-number field. If this is not the case, instead of eq. (5.1) one has a similar one, but with E, B and f being no longer c-number functions, but quantum mechanical operators. This equation is then of course coupled to the system of Maxwell equations. To show how the Wigner function approach works in this case, let us investigate an atom coupled to an electromagnetic field”). The Hamiltonian for an atom with one optically sensitive electron reads:
ii=&: [ii-fa(Ri,r,]2+$: [~.+~‘qRJ)]‘: I
+
O(R, - Ri) + e[ d(RiYt) -
e
&Re,f)l .
(8.1)
Quantities with index “i” describe here a positively charged core and those with index “e” describe an electron. The density matrix evolution equation gives the following equation for the gauge independent Wigner function:
+~[-VoQ,-Ri)].(~-~)))(R,.P,;Ri,P,;~):=O. I
(8.2)
GAUGE INDEPENDENT WIGNER FUNGI-ION APPROACH
583
Here :( ): means normal ordering, i.e. ordering where all the electromagnetic field creation operators are on the left-hand side of particle operators and annihilation operators are respectively on the right-hand side. Eq. (8.2) is a direct two particle generalization of (2.2). The new term &[-VO(R,-RJ]
:= [-V O(ReARJ]
[t ” (&-$)I-’ I
Xsin[tV*(k-k)]
is responsible for the mutual electron-ion interaction. In the long wave (semiclassical) limit we can put ti[F(Ri,,?
t)]
=
F(Ri,,T
r>,
61i.e=Pi.elM,e’
Leaving only the terms of the first order in Pi,e/Mi,.C and in spatial derivatives of fields (dipole approximation) and neglecting Me/M, in comparison with 1 we obtain from (8.2)
:
1$i..V,+ni[-V.O(r)].$
Here lower case letters describe the relative motion of the electron and the positively charged core whereas the capital ones describe the center of mass motion. If we now describe relative motion in the representation given by the atom’s stationary states, from (8.3) we get the following equation:
584
P. BADZIAG
We have neglected functions
faD are defined
fmB = &(R,
P; t) :=
r
u,(r)
is the
with E. The
as follows:
d3p d3r, d3r, exp
X r+(r,)*u,(r,)!(ir, where
x B and (rp - V) - E in comparison
here (l/c)V
+ $r2,p;
electron
wave
fP.(r,-rJ
I
R, P; r), function
in the
stationary
state
la)
and
:= (a/i//?>. @Eq. (8.4) is directly
obtained
from (8.3) after multiplying
both sides of (8.3)
by
up(rI)*um(r2) ev i and then substituting Eq. (8.4) describes
:P*(rl - r2>I
i(r, k rJ for r and finally integrating over d”p d3r, d3r,. both emission and absorption of light as well as radiation
pressure. In this article we focus on the interaction between ion) motion and electromagnetic field, leaving the problem sure for the next publication. Without
the radiation
pressure
f$(R,P;t)+ie:&R,
terms
relative (electronof radiation pres-
(8.4) reads:
t)*[r,,&(R,
P;t)-rPvfa,,(R,P;
t)]:=O. (8.5)
This is coupled
to the system
Hereafter we put e = r? = 1. In the dipole approximation
j
of Maxwell
equations
of the two-level
CR,f) = g woh(R)12 + lu2(R)121[ r2,h -
atom one has
r,,f,,l
P(R f>= e[u,(R)f,,ui(R) + ~~(R)f~~u:(R)l. If we put here
which give
I
(8.7a)
(8.7b)
GAUGE INDEPENDENT WIGNER FUNCIION APPROACH
58.5
assuming the atom under consideration being localised in R = 0. Eq. (8.6) together with (8.7a, b) and the decomposition of a transverse part of the electric field
give the following equation for i2, operators:
Eqs. (8.5) and (8.8) are slightly different from similar ones obtained on the basis of the Hamiltonian approach (no matter whether it is a minimal coupling Hamiltonian”) or a Hamiltonian of Power and Zienau**)), but they lead to the proper expressions for the emission line width as well as for the Lamb shift. It is just simple algebra to show that in the Markov approximation (8.5) and (8.8) give 1
4
ku
h
1
P--k-o,
1 k+w,
1 .
9. Concluding remarks The Wigner function approach to the quantum kinetic problems we present in this paper enables a relatively unique description of Fermi as well as Bose particle systems; however, different approximations at the final stage of calculations, in particular different approaches to the self-consistent field approximation, have to be done. This problem is illustrated by the comparison of the calculations of dispersion relations of excitations in the system of Fermiand Bose-particles. We would also point out that because of the gauge independence of the formalism, we use throughout the paper, equations we are dealing with are free from unmeasurable quantities such as electromagnetic potentials, even when they describe charged particles in an electromagnetic
586
P. BADZIAG
field. The last statement quantized. enabled
In
this
motion
case
us to avoid
the polarization
the
gauge
independent
both electromagnetic
term
describing
also refers to the case when the electromagnetic
in the Power
spontaneously
and
radiating
Wigner
potentials Zienau
function
and divergent
approach**))
field is approach
terms
(like
in equations
of
atoms.
Acknowledgement I thank
Professor
L.A. Turski
for several
stimulating
and fruitful
discussions.
References 1) 2) 3) 4) 5) 6) 7) 8)
H. Weyl, Z. Phys. 44 (1927) 1. E.P. Wigner, Phys. Rev. 40 (1932) 729. E.A. Remler, Ann Phys. 95 (1975) 455. P. Carruters and F. Zachariassen, Rev. Mod. Phys. 55 (1983) 245. P. Carruters and F. Zachariassen, Phys. Rev. D 13 (1976) 950. Ch. C. van Weert, W.P.H. de Boer, Physica 81A (1976) 597, 85A (1976) 566, 86A (1977) 67. L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1%2). S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics (Saunders Philadel-
phia, 1966), p. 75. 9) R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, 1975). V.P. Sihn, Vvedenie v Kinetitcheskuiu Teoriu Gazov (in Russian), Moskva (1971). 10) W.E. Brittin and W.R. Chappell, Rev. Mod. Phys. 34 (1962) 620. 11) E.A. Remler, Phys. Rev. D 16 (1977) 3464. U. Heinz, Phys. Rev. Lett. 51 (1983) 351. 12) For the detailed presentation of the properties of the Wigner function, the reader is referred to S.R. de Groot and L.G. Suttorp, Foundations of Electrodynamics (North-Holland, Amsterdam 1972) as well as to the recent review article of N.L. Balazs and B.K. Jennings, Phys. Rep. 104 (1984) 347, where in particular one can find deeper insight into the relation between different definitions of the Wigner function (like for instance (2.1) and (3.1) in our article). 13) R.J. Glauber. Phys. Rev. 130 (1%3) 2529, 131 (1963) 2766. 14) L.A. Tutski, Physica 57 (1972) 432. 15) E.P. Gross, J. Math. Phys. 4 (1%3) 195. 16) The first proof of this kind, but of a weaker statement (instead of LY*@2 aft’, o*p* 3 ift* was necessary to prove it) was given by N.D. Cartwright, Physica 83A (1976) 210. Lately we have also found a proof like ours in N.W. Bakers and B.K. Jennings, Phys. Rep. 104 (1984) 347. 17) The problem is related to the gauge independent formulation of quantum electrodynamics, where instead of the freedom to choose the particular gauge, one has the freedom to choose the form of the compensating current (cf. I. Bialynicki-Birula and L. Bialynicka-Birula, Quantum Electrodynamics, (Pergamon, London, 1975), pp. 352-359). 18) A.S. Davydov, Kvantovaja Mechanika (in Russian) (Fizmatgiz, Moscow 1%3), p. 676. 19) L. Allen and J.H. Eberly, Optical Resonance and Two-level Atoms, (Wiley, New York, 1975) chap. 7. 20) S. Stenhohn, Phys. Rep. 43 (1978) 151. 21) V.S. Letokhov and V.G. Minogin, Phys. Rep. 73 (1981) 1. 22) E.A. Power and S. Zienau, Phil. Trans. R. Sot. A251 (1959) 427.
GAUGE
INDEPENDENT
SPONTANEOUS
WIGNER FUNCTION APPROACH
EMISSION AND TO QUANTUM
KINETIC
TO
EQUATIONS
Piotr BADZIAG of Physics, University of South Africa, P 0 Box 392, Pretoria, South Africa
Deparhnent
Received 5 July 1984 Revised 4 October 1984
Equations for the evolution of the gauge independent analogue of the Wigner function are derived and applied to the description of a many charged particle system. The self-consistent field approximation is studied more closely. Possible applications of the formalism are illustrated by the calculations of dispersion relations for elementary excitations of fermion as well as boson systems. A possibility of application of the formalism in quantum optics is also investigated.
1. Introduction
The Weyl-Wigner formulation of quantum mechanics’,*) has been widely used in different fields of physics. There are well-known Wigner formulations of scattering theory3*4), relativistic quantum mechanics5*6) and quantum kinetic theory7-9) as well as some approaches via the Wigner representation to quantum electrodynamics”) and quantum field theory”). Although the Wigner representation is not commonly used in above-mentioned fields because of the complexity of the equations, in most of the applications it is very useful in the investigations of the long wave phenomena directly connected to the semiclassical limit. In this limit the Wigner function becomes the classical phase space density, whereas describing charged particles in an electromagnetic field neither Wigner function nor phase space density are real objects,
since they are not gauge independent.One
can slightly
change
the
definition of the Wigner function to make it gauge independent and to give it an interpretation of the quantum analogue of the classical density in the position-kinetic momenta space’,“). Th e main purpose of this paper is to apply the gauge independent Wigner function formalism to the kinetic theory of charged particle gas as well as to the description of the interaction between atoms and quantized electromagnetic field. When dealing with kinetic equations we restrict ourselves to the spin f and spinless particles and finally 0378-4371/85/$03.30 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
P. BADZIAG
566
concentrate on the self-consistent field approximation. gives elementary definitions, examples and theorems representation’).
In section
charged particle We also present
3 we redefine
Section connected
the Wigner
2 of this paper to the Wigner
function
describing
a
in an electro-magnetic field to make it gauge independent**“). some properties of the gauge independent Wigner function.
This section
introduces
also the generalization
ning particle
case. In section
4 we begin
of the formalism
with the description
for the spin-
of many
particle
systems: introduce a concept of reduced gauge independent Wigner functions and list some properties of them. Section 5 gives the discussion of the quantum Vlasov approximation and comparison of the approximation with the HartreeFock one. Sections 6 and 7 give a discussion of the linearized Vlasov equation describing fermion and boson systems, respectively. The important role of the coherent states13) in the case of Bose particles becomes here visible’4X’5). Our approach to derive the Landau criterion for superfluidity can be in this context compared with the one of Gross”); however, do not need to assume that the excitations
to derive the proper formula, we are long wave ones, what Gross
actually needed (linearization not only in the density variations 6p but also in the variations of the phase of the wave function 8s). Finally, in section 8 we generalize the formalism to the case when the electromagnetic field is quantized and show that this generalization enables a straightforward description of atoms interaction with a quantum electromagnetic field. Some concluding remarks are presented in section 9.
2. Wigner function The Wigner
function
matrix
in coordinate
particle
its Wigner
is usually
defined
representation. representation
as a Fourier
When is defined
density
transform matrix
of the density
fi describes
one
as follows:
(2.1) The function defined in this way is the quantum analogue of the classical phase space density. To justify this statement let us enumerate some of the properties of the Wigner function”). Expressions for the coordinate space probability density, momentum space probability density and current density are given by classical relations:
i)
p(R, f) = j
d”pf,(R
P;
1)2 0,
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
ii>
iii)
P(P,t) = j d3Rfw(R P; t) 2 0 , j (R, t) = j dp tf,(R,
p; t) .
If 8 is the operator which represents an observable, then the expectation of 8 in the state described by fw( .) is given by
iv)
561
(6) = 1d3pdZRO,(R, p)f,(R,
value
p; 0.
Here O&t, P; t) =
1d3r (R -
kr18(R + it-) exp (ip
- r)
is the Weyl transform of the 8 operator and is closely related to the classical observable. One can easily show for example that
W%(R
P)
= f(p),
[f(@l,(RP) =f(R).
Of course one cannot prove that same for arbitrary f(k, $), because k and a do not commute. Although the Wigner function itself is real, it is not the density in the classical meaning, since it is not non-negative. This is connected with the uncertainty principle, so that the quantum analogue of the positive definiteness of the classical distribution function reads:
v)
(27r~rp)-~ 1 d3p d3Rfw(R, p) exp [ - w
- (RiPF)2]
2 0.
To prove (v)‘“) note that for two density matrices
(*I In the Wigner representation
(*) reads:
568
P. BADZIAG
Thus,
to obtain
(v) we have only to write
where
h2
(y2=-
4p2. f2 defined above is the Wigner function packet given by the wave function
i+b(R) = (27@2)-3’2exp
The Wigner representation quite similar to the classical
[
- (R4B?)2]
which
describes
the Gaussian
wave
exp (- i pO - R)
of the Liouville-von one. For a particle
Neuman in external
equation potential
has a form field Q(R)
this reads
(2.2) Here F(R)
= -V@(R)
1
112
d3r d3q
=
(2n-h)3
exp
dA F(R
[-ir*(P-cl)
+ Ar)
l/2
i =-Ih
d3r d2q (277-h)3 exp
I
-~r*(p-q)].[@(R+~r)-@(R-;r)]fw(R,q).
stfw(R.
q)
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
569
To make the semiclassical limit of the integral operation &[ F(R)] more visible one can write it in the form of the formal series of the differential operators:
One can easily see now that in the limit when F(R) is a slowly varying function eq. (2.2) gives the classical Liouville equation. One can also see that in the case of F being linearly varying (constant force, harmonic oscillator) (2.2) is exactly the same as its classical counterpart.
3. Charged particle in an electromagnetic
field
If we consider a charged particle in an electromagnetic field, eq. (2.2) is no longer so transparent. One of the reasons for it is that &(R,p; t), just like the classical phase space distribution function, depends on the choice of gauge of electromagnetic potentials. Therefore in classical theory one often uses the position-velocity space distribution instead of the phase space one. Following this idea we modify the definition of the Wigner function to make it a quantum analogue of the classical position-kinetic momentum density. There are of course many different possibilities for such a modification”). To make our choice more unique let us rewrite definition (2.1) in the form:
(3.1)
Simple, but a little tedious algebra shows that (3.1) and (2.1) are indeed equivalent. By substituting canonical momentum operator @ by the kinetic momentum one, P - (e/c)A in (3.1) we get (compare refs. 8 and 11). l/2
exp{-ir.[p+E
1
dAA(R+hr,t)]]
-l/Z
xp(R+;r,R-;r;
t).
(3.2)
570
P. BADZIAG
One
can easily
and
has properties
position
verify
and kinetic
any observable
that the function
of the quantum momentum
defined analogue
space.
by (3.2) is gauge of the classical
In particular
independent density
the expectation
in the value
of
8 is given by:
(6) = j- d3Rd3pO(R, p)f(R P)
(3.3)
1
where O(R,p) is the Weyl transform of the 0 operator modified in a similar way to the Wigner function. For the one particle operator it reads:
O(R,p)=ld’r(R-;r@jR+;r) l/2
xexp(kr*[p+:
1
dAA(R+hr,f)]].
(3.4)
-l/2
Thus, for instance, reads :
the probability
density
of finding
a particle
in d3R around
AR, 0 = (IJWRI)= j- d3pf(RP; f) ; the current
R
(3.5)
density:
(3.6)
Different coordinates of the kinetic momentum operator do not commute, so that one cannot expect that property (v) of the Wigner function transforms directly into the gauge independent function. There is, however, an analogue of that property which, at least in the case when the magnetic field is uniform, has a direct and transparent form -namely: for &P+P, 1 f&R,,
~0; t) = ~2Tap~~
x exp _
I
1
d3p d3R
f (R, p; t) exp
-
(R - R,)* w2
b
-PO
+
I.
(42cP x CR- Ro)12 , 2CX2
1
o
(3.7)
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
571
The proof of (3.7) is just the same as that of (v) in section 2. It is only to be noted that the pure state described in the Coulomb gauge by the wave function
JI(R)
=
Q+*)-~‘*
exp
in the gauge independent
-
(Rip?‘2 +f R * (p.+5 B x R,,)]
Wigner representation
f2M P) = (27~Lyp)-~ exp{ -
(R - Ro)* _ [P -PO
is given by +
W2c)Bx (R -
2p2
2cu2
Ro)l*
,
I
where (y*=-
h2 4p* *
It is worthwhile to point out, that f*(a) d escribes a state in which position and kinetic momentum coordinates perpendicular to B are uncertain up to 2 u ,l
-p*,
e2B2p2
u;,=a*+4cZ.
3.1. Evolution of the gauge independent Wigner function It is just a matter of a simple but quite tedious algebra to show that the Liouvelle-von Neuman equation in the gauge independent Wigner representation reads: l$tL11_P+e(ti[E(R,t)]+fL;,x~[B(R,t)])+}f(R,p;t)=0.
Here fi[F]
is described in section 2; B,f=$[pf(R,p)+~~~
7 dA exp(fr.q)[B(R-hr)+hr]) -l/2
xf(R,p-4)
(3.8)
572
P. BADZIAG
In the limit when E and B are slowly varying for the evolution of the probability space. Here, like in the orthodox
density Wigner
forces, (3.8) is the same as its classical and B is uniform. 3.2. Spinning
(3.8) gives the classical in coordinate-kinetic representation and
limit when E changes
equation
momentum the potential
linearily
in space
particle
Eq. (3.8) describes the particle possesses nonzero
evolution of the spinless spin the density operator
particle only. in coordinate
When the represen-
tation is the matrix function paP(R,, R,;t).where (Y.p are spin indices. This indicates that instead of introducing only one Wigner function we have to introduce the matrix of functions: faa( R,p; t). This makes the whole formalism more complicated; however, at least for spin i particles. all linear independent functions have a clear physical interpretation, namely
describes
the quantum
analogue
of probability
density.
with u the Pauli matrices, describes 2/h times the spin density. For the time being we restrict ourselves to this simple case. In this case, instead of (3.8) we have
+ ~~[B(R,t)](6.~~.s(R,P:t)=O.
(3.9a)
(3.9b)
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
573
One can easily see that when B is uniform integration of (3.9b) over the momentum space gives the well known equation for spin precession:
(3.10)
~s.(Qf)+vj~j;(R,l)+~EiPBISI(Qf)=O. Here si(R, r> = 1 d3Psi(R, p; t) describes the space distribution
of spin
Tji(R,t)=Idip~p,si(R,p;t); which describes the spin current.
4. Many particle system To describe a many particle system one has to work with the reduced density matrices. The same applies to the Wigner formalism. We define the one particle reduced gauge independent Wigner function for spin zero particles as follows: l/2
3 (:exp[-ii*[p+: h(R~Wj&y
1 d,iA(R+Ar,l)]} -l/2
x I&R -
;r,t)t,i(R + ;r):).
(4.1)
Here 4 are the field operators, :( ): is the symbol of normal ordering and ( ) is the symbol of the expectation value. The generalisation of (4.1) on the spinning particles is straightforward. We only have to add spin indices to fi, which now is the matrix fld and to 4’ and 4 which now become 4: and &. For spin l particles instead off,& we can choose the following linear independent combinations of these functions:
f,(R,p; O= ~fl,za(R~; 6, n
(4.2a)
P BADZIAG
574
s,(R
Pi
t) = c u,,f,,(Rp;t)
(4.2b)
nb
Here &‘s are the Pauli matrices. The many particle reduced defined
in an analogous
fzabcAR,, PI; R,,
gauge
independent
way, i.e. two particle
Wigner
functions
functions
are
are
PS t) 112
1 dAA(R,+Ar,,l)]}
=]$$$(:exp{-iir,-[ktf
-112 x I&R, - ;r,, t)&(R, Commutation relations between conditions for f2 functions: f2,,‘&R,,
~1;
R,,
~2;
f> =
- 1 ?r2, t)x ~~(R,+~r,,t)~b(R,+~r,, field operators
fwati&
P2; RI,
yield
the following
t): symmetry
(4.4a)
PI; f, 3
+w,.(R,--R,-~r,-fr2)+w2~(RZ-R,-~rl-~r2)+~
c 1 d* S - Bk
=: *&2f2abcd(R,t PI; R,, P2; t). Here
0
is the surface
f)] ]
(4.4b)
of the quadrangle
bounded
by the sectors
between
the
points R,-ir,,
R,+ir,,
R,+ir,,
R,-kr,.
The order of these points and the right-hand screw rule give direction of the surface, the element of which is denoted by d*S.
5. Evolution
of f,
Particles
interact
with each other,
so that the evolution
equation
the
for
positive
fi
is no
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
longer as simple as (3.8) or (3.9). If the interaction is via the potential with potential U(jRI), we get the following equation for f,:
d3R’d3p’lk[VU(~R-R’~)]~$f,,,(R,p;R’,~’;~).
=-
515
force
(5.1)
Here p denotes the gyromagnetic ratio, ii are the spin divided by h operators. One can rewrite the right-hand side of (5.1) in a more evident form: d3q d3r d3R’ d p (2d)3 exp 3
&
= f
,
’
x[U(jR-R’+;tl)-U(IR-R’-;rl)]f2dcc(R,q;R’,p’;f).
(5la)
Equations for f, involve f3 and so on, and thus we obtain the quantum analogue of classical BBGKY hierarchy. This is, however, still a hierarchy of classical character in a sense that the electromagnetic field is considered here as a given c-number field. If it is not the case, instead of products of the type
we
have
:@(R, th(R, where k ordering. going to imation)
P;
t): ,
and f& are quantum mechanical operators and :( ): denotes normal We will return to this problem in section 8 and for the present we are investigate the self-consistent field approximation (Vlasov approxof eq. (5.1).
576
P. BADZIAG
The
classical
Vlasov
q; t)*flcc(R’,p’;
f,,(R,
does not, however, quantum
theory
approximation t) instead
preserve
is obtained,
when
in (5.la)
one
puts
of fzabcc(R, q; R’, p’; t). This approximation
the proper
symmetry
of fi. Because
of this, in the
one has to put rather (5.2)
Here
the upper
proper
sign refers
self-consistent
to bosons
and the lower
field approximation
fmcc(R 4; R’, P'; f)==#+ &f,,(R’,p’;
for bosons t)f,JR,
to fermions.
Actually
the
is rather q;
r) .
In section 7 we discuss this problem in more detail. To see the quantum corrections to the classical Vlasov equation let us examine the quantum Vlasov approximation of eq. (5.1). We will do it separately for fermions and separately for bosons since the natures of the ground state for both types of particles are different, ergo the different types of approximation
are useful.
6. Elementary
excitations - fermions
For the sake of simplicity we restrict ourselves to spin i particles. In this case we can rewrite the quantum Vlasov approximation of (5.1) in the form of coupled equations for f, and s, defined by (4.2a) and (4.2b), respectively:
(6.1 a)
(&lb)
Here
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
d3q d3r
R,(R,p;r)=$/d3R’d3p’----
Lr*(p-q)][U(lR-R’+;rl) h
(27rh)3 exp
h)f,(R
WIR - R’ - ;rl)l(l -
4;
t)f,(R’, P’; 9, &p-q)
-
U(l R - R’ - ;rj)](l
511
I
[~(IR-~f+;rl)
- &)s,(R, q; t)f,(R’, P’; t) .
In the case of the magnetic field being not too strong, i.e. eiiB -4~: C
for low temperatures,
or eflB
-4
C
2mk,T
for the high temperature
limit,
we can neglect the magnetic field in the definition (4.4b) of the p,* operation, so that in this case we can write
B,,[~,(R’,P’)~(R,‘I)I=~~~~~~(~[~.(R-R’)-(P-~).~IJ R+R’-r
p’+q+T
2
)
2
R+R’+r >fi (
2
p+q-7 ’
2
)
(6.2a)
~~~[f,(R’,p’)s,(R,q)l~~~~exp{~~~.(R-R’)-(~’-q).rl} R+R’-r 2 +
(s1*f1>-
p’+q+T )
%,f,
R+R’+r 2
2
--, s, x Sl> . 1
p+q-7 ’
2
>
(6.2b)
This approximation makes (6.la) and (6.lb) even simpler than their gauge dependent analogue, known e.g. from Silin’s or Balescu’s textbooks. The
578
P. BADZIAG
information equations
contained
our approach wave
in (6.1) is of course
for the phase
space Wigner
the quantities
limit,
a straightforward,
the absence
of the magnetic
the same as the one contained
function.
that we are dealing classical,
gauge
The main with have, independent
field our equations
difference
in the
is that in
at least in the long interpretation.
are the same as those
In
for the
phase space Wigner function and after linearization describe the longitudinal plasma oscillations (when fO is the Maxwell distribution) as well as the excitations of the Fermi liquid (when fO is the Fermi-Dirac distribution and describes quasiparticles rather than particles). The crucial role of the exchange term becomes apparent in the later case. We are, however, not going to proceed along this line, because the results are well known and the reader can find them in textbooks like those by Silin or by Balescu’).
7. Elementary
excitations - bosons
In this section we are going to investigate the dispersion relations for the elementary excitations in a system of weakly interacting Bose particles in the low temperature limit. For the sake of simplicity we restrict ourselves to the case of spinless particles and no external magnetic field present. The quantum Vlasov approximation of eq. (5.1) now reads:
x ;[U(jR
-R’+;rj)-
U/(/R - R’-;rl)](l
+ &f,(R,
q; f)f,(R’,p’;
t).
(7.1) To derive
(7.1) let us have
a look
at the
derivation
of the
Hartree-Fock
approximation. The basic assumption which leads to this approximation all the particles (quasiparticles) are in different one-particle states, N-particle state is of the form
I$,)= ii;, . . . a;p>, where
all k,, . . . , k,
is that i.e. an
(7.2)
are different.
Here I@) is the vacuum state and (~2:~) are creation operators the states {ki}. Function fi( * ) is the Fourier transform of the expectation product:
of particles value
in
of the
GAUGE INDEPENDENT WIGNER FUNCTION APPROACH
The expectation
519
value of (7.3) in the state (7.2) is
The last equality leads directly to
When we deal with Bose particles, states like the one described by (7.2) are of no value, because in this case many particles can occupy the same quantum state. Instead of the states of the type (7.2) the coherent states are distinguished now (at least in the problems connected to superfluidity). Coherent states are the eigenstates of anihilation operators, i.e. (7.5) This leads to
We choose the last form, because it preserves the proper symmetry of the interaction term in the evolution equation. The last equality in (7.6) is equivalent to
f*( - ) =
a+ kM(
-)f*( - )
7
which we use in (7.1). For spinless particles and no magnetic field we have
(7.7)
P. BADZIAG
580
Xf,(;[R+ RIf we now collect
r],;[p’+
q + ~l)f,(;[R+
R’+ r],;[p’+q-
T]).
(7.8)
(7.1) and (7.8), then substitute
f,(R, P; t) = A,(P) + V(R> P; t) and leave only the linear
X
the spin zero analogue
of (6.3):
exp [-~(R-R’).(p-q)j-ti(2ip-qj)~~p{$R-R’)~(p-q)]j
x { Cf(p’Pf(R +
terms in Sf, we obtain
d3r d37 (2d)”
exp
t) + fo(qPf(R’, p’; f)l
4;
{&(R-R’)-(p’-q)v]} h
x [f&p’+ q + dPf(;[R + R’+ X 6f(i[R
+ R’-
r], l[p’+
q +
T];
r],f[p’+ r)]}.
q -
71;t>+fo&~'+
9-
71) (7.9)
When
6f(R,P; f>= exp {f (k - R -
4}f,(p)
eq. (7.9) gives
i
k+pJm -w+l
=
1 d3q ol(l~ -
ql)[fo(q+ ;k) -fob - %)I\fdd
;[fo(p+ ;k)-fob - ;kl j d3q[~#I)+
When the interaction most of the particles
ol(lp
-
ql)lfk(q).
is weak and repulsive, in the ground are condensated, so we can write
(7.10)
state of the system
GAUGE
INDEPENDENT
WIGNER
FUNCTION
APPROACH
581
h(P) = %6(P). This gives the final form of the equation for fk(p):
= ;~,P(P + $4 - S(P- ;k)lI d3q[~(IW + ~I(IP- ~l)ltX~).
(7.11)
Because of the delta functions on the right-hand side of (7.11) we can get get dispersion relation from this equation without making any additional approximations. We can rewrite (7.11) in the form
= ;@(P
-
;@(P
+
I
0th + ;kl)]fk(q) ;k) d3q[ fiI(lkl)+
-;k)I d3q[fil(lkl)+ oI(lq- ~kl>l.h(4)
(7.12)
3
or ;r@(p ‘(‘)
+ ;k)
= -k*/(2m) - w + in,,[ l_?(O)- e(jkl)] ;n,S(p
xk
- ;k)
- k2/(2m) - o + $n,,[o(lkj) - a(O)] yk ’
(7.12’)
where by X, and Yk we have denoted the following integrals:
xk
=
yk = Integration
d3q[~l(lkl)+fil(bl+;kl)l_fi(‘d 3
d3q[~(lkl) + ol(i’I- ;kl)lf,(‘d. over d3p of the both sides of (7.12’) multiplied by
1) {o(M) + ti(I, + ;kl)L
2)
@l(lkl) + o(lp - %I)1
gives the linear and homogeneous set of two equations solvability condition for this set reads:
for X, and Yk. The
P. BADZIAG
582
w2
_
_
k2
I 24W*
(2m)’
2m
(7.13)
’
where v(k) = n,,~(~k~) The relation (7.13) obtained from the proper self-consistent field proximation (7.1) is just the same as the one obtained from diagonalization the many particle Hamiltonian by the Bogolubov transformation method e.g. ref. 18) and because of the coherent states, which are the basis of self-consistent field approximation, it is related to the same result obtained GrossIS) from the hydrodynamical form of quantum mechanics.
8. Atoms in a quantized
electromagnetic
apof (of our by
field
So far we have been investigating systems, where the electromagnetic field was a c-number field. If this is not the case, instead of eq. (5.1) one has a similar one, but with E, B and f being no longer c-number functions, but quantum mechanical operators. This equation is then of course coupled to the system of Maxwell equations. To show how the Wigner function approach works in this case, let us investigate an atom coupled to an electromagnetic field”). The Hamiltonian for an atom with one optically sensitive electron reads:
ii=&: [ii-fa(Ri,r,]2+$: [~.+~‘qRJ)]‘: I
+
O(R, - Ri) + e[ d(RiYt) -
e
&Re,f)l .
(8.1)
Quantities with index “i” describe here a positively charged core and those with index “e” describe an electron. The density matrix evolution equation gives the following equation for the gauge independent Wigner function:
+~[-VoQ,-Ri)].(~-~)))(R,.P,;Ri,P,;~):=O. I
(8.2)
GAUGE INDEPENDENT WIGNER FUNGI-ION APPROACH
583
Here :( ): means normal ordering, i.e. ordering where all the electromagnetic field creation operators are on the left-hand side of particle operators and annihilation operators are respectively on the right-hand side. Eq. (8.2) is a direct two particle generalization of (2.2). The new term &[-VO(R,-RJ]
:= [-V O(ReARJ]
[t ” (&-$)I-’ I
Xsin[tV*(k-k)]
is responsible for the mutual electron-ion interaction. In the long wave (semiclassical) limit we can put ti[F(Ri,,?
t)]
=
F(Ri,,T
r>,
61i.e=Pi.elM,e’
Leaving only the terms of the first order in Pi,e/Mi,.C and in spatial derivatives of fields (dipole approximation) and neglecting Me/M, in comparison with 1 we obtain from (8.2)
:
1$i..V,+ni[-V.O(r)].$
Here lower case letters describe the relative motion of the electron and the positively charged core whereas the capital ones describe the center of mass motion. If we now describe relative motion in the representation given by the atom’s stationary states, from (8.3) we get the following equation:
584
P. BADZIAG
We have neglected functions
faD are defined
fmB = &(R,
P; t) :=
r
u,(r)
is the
with E. The
as follows:
d3p d3r, d3r, exp
X r+(r,)*u,(r,)!(ir, where
x B and (rp - V) - E in comparison
here (l/c)V
+ $r2,p;
electron
wave
fP.(r,-rJ
I
R, P; r), function
in the
stationary
state
la)
and
:= (a/i//?>. @Eq. (8.4) is directly
obtained
from (8.3) after multiplying
both sides of (8.3)
by
up(rI)*um(r2) ev i and then substituting Eq. (8.4) describes
:P*(rl - r2>I
i(r, k rJ for r and finally integrating over d”p d3r, d3r,. both emission and absorption of light as well as radiation
pressure. In this article we focus on the interaction between ion) motion and electromagnetic field, leaving the problem sure for the next publication. Without
the radiation
pressure
f$(R,P;t)+ie:&R,
terms
relative (electronof radiation pres-
(8.4) reads:
t)*[r,,&(R,
P;t)-rPvfa,,(R,P;
t)]:=O. (8.5)
This is coupled
to the system
Hereafter we put e = r? = 1. In the dipole approximation
j
of Maxwell
equations
of the two-level
CR,f) = g woh(R)12 + lu2(R)121[ r2,h -
atom one has
r,,f,,l
P(R f>= e[u,(R)f,,ui(R) + ~~(R)f~~u:(R)l. If we put here
which give
I
(8.7a)
(8.7b)
GAUGE INDEPENDENT WIGNER FUNCIION APPROACH
58.5
assuming the atom under consideration being localised in R = 0. Eq. (8.6) together with (8.7a, b) and the decomposition of a transverse part of the electric field
give the following equation for i2, operators:
Eqs. (8.5) and (8.8) are slightly different from similar ones obtained on the basis of the Hamiltonian approach (no matter whether it is a minimal coupling Hamiltonian”) or a Hamiltonian of Power and Zienau**)), but they lead to the proper expressions for the emission line width as well as for the Lamb shift. It is just simple algebra to show that in the Markov approximation (8.5) and (8.8) give 1
4
ku
h
1
P--k-o,
1 k+w,
1 .
9. Concluding remarks The Wigner function approach to the quantum kinetic problems we present in this paper enables a relatively unique description of Fermi as well as Bose particle systems; however, different approximations at the final stage of calculations, in particular different approaches to the self-consistent field approximation, have to be done. This problem is illustrated by the comparison of the calculations of dispersion relations of excitations in the system of Fermiand Bose-particles. We would also point out that because of the gauge independence of the formalism, we use throughout the paper, equations we are dealing with are free from unmeasurable quantities such as electromagnetic potentials, even when they describe charged particles in an electromagnetic
586
P. BADZIAG
field. The last statement quantized. enabled
In
this
motion
case
us to avoid
the polarization
the
gauge
independent
both electromagnetic
term
describing
also refers to the case when the electromagnetic
in the Power
spontaneously
and
radiating
Wigner
potentials Zienau
function
and divergent
approach**))
field is approach
terms
(like
in equations
of
atoms.
Acknowledgement I thank
Professor
L.A. Turski
for several
stimulating
and fruitful
discussions.
References 1) 2) 3) 4) 5) 6) 7) 8)
H. Weyl, Z. Phys. 44 (1927) 1. E.P. Wigner, Phys. Rev. 40 (1932) 729. E.A. Remler, Ann Phys. 95 (1975) 455. P. Carruters and F. Zachariassen, Rev. Mod. Phys. 55 (1983) 245. P. Carruters and F. Zachariassen, Phys. Rev. D 13 (1976) 950. Ch. C. van Weert, W.P.H. de Boer, Physica 81A (1976) 597, 85A (1976) 566, 86A (1977) 67. L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1%2). S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics (Saunders Philadel-
phia, 1966), p. 75. 9) R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, 1975). V.P. Sihn, Vvedenie v Kinetitcheskuiu Teoriu Gazov (in Russian), Moskva (1971). 10) W.E. Brittin and W.R. Chappell, Rev. Mod. Phys. 34 (1962) 620. 11) E.A. Remler, Phys. Rev. D 16 (1977) 3464. U. Heinz, Phys. Rev. Lett. 51 (1983) 351. 12) For the detailed presentation of the properties of the Wigner function, the reader is referred to S.R. de Groot and L.G. Suttorp, Foundations of Electrodynamics (North-Holland, Amsterdam 1972) as well as to the recent review article of N.L. Balazs and B.K. Jennings, Phys. Rep. 104 (1984) 347, where in particular one can find deeper insight into the relation between different definitions of the Wigner function (like for instance (2.1) and (3.1) in our article). 13) R.J. Glauber. Phys. Rev. 130 (1%3) 2529, 131 (1963) 2766. 14) L.A. Tutski, Physica 57 (1972) 432. 15) E.P. Gross, J. Math. Phys. 4 (1%3) 195. 16) The first proof of this kind, but of a weaker statement (instead of LY*@2 aft’, o*p* 3 ift* was necessary to prove it) was given by N.D. Cartwright, Physica 83A (1976) 210. Lately we have also found a proof like ours in N.W. Bakers and B.K. Jennings, Phys. Rep. 104 (1984) 347. 17) The problem is related to the gauge independent formulation of quantum electrodynamics, where instead of the freedom to choose the particular gauge, one has the freedom to choose the form of the compensating current (cf. I. Bialynicki-Birula and L. Bialynicka-Birula, Quantum Electrodynamics, (Pergamon, London, 1975), pp. 352-359). 18) A.S. Davydov, Kvantovaja Mechanika (in Russian) (Fizmatgiz, Moscow 1%3), p. 676. 19) L. Allen and J.H. Eberly, Optical Resonance and Two-level Atoms, (Wiley, New York, 1975) chap. 7. 20) S. Stenhohn, Phys. Rep. 43 (1978) 151. 21) V.S. Letokhov and V.G. Minogin, Phys. Rep. 73 (1981) 1. 22) E.A. Power and S. Zienau, Phil. Trans. R. Sot. A251 (1959) 427.